Area of Trapezium
The area of a trapezium is defined as the number of unit squares that can fit into it and it is measured in square units (like cm2, m2, in2, etc). A trapezium is a type of quadrilateral that has one pair of parallel sides(generally referred to as bases). The other pair of sides of a trapezium can be non-parallel and are known as legs. The area of a trapezium is the total space covered by a trapezium in a two-dimensional plane. For example, if 20 unit squares each of length 1 in can be fit inside a trapezium, then its area is 20 in2. It is not always possible to draw unit squares and hence measure the area of a trapezium. We will learn about the area of a trapezium and the formula to calculate it on this page.
|1.||Area of Trapezium Formula|
|2.||How to Derive Area of Trapezium Formula?|
|3.||How To Calculate Area of Trapezium?|
|4.||FAQs on Area of Trapezium|
Area of Trapezium Formula
The area of a trapezium can be calculated using the lengths of two of its parallel sides and the distance (height) between them. The formula to calculate the area (A) of a trapezium using base and height is given as,
A = ½ (a + b) h where,
- a and b = bases of trapezium, and,
- h = height (the perpendicular distance between a and b)
How to Derive Area of Trapezium Formula?
We can derive the formula to find the area of a trapezium in two ways:
- Using a parallelogram
- Using a triangle
Derivation of Area of Trapezium Formula Using a Parallelogram
To derive the formula for the area of a trapezium using parallelogram, we will consider two identical trapeziums, each with bases a and b and height h. Let A be the area of each trapezium. Assume that the second trapezium is turned upside down as shown in the figure below.
Now, join the above two trapeziums.
We can see that the new figure obtained by joining the two trapeziums is a parallelogram whose base is a + b and whose height is h. We know that the area of a parallelogram is base × height. The area of the above parallelogram is, A + A = 2A.
Thus, 2A = (a + b) h
⇒ A = (a+b)h/2
Thus, the formula for the area of a trapezium is derived.
Derivation of Area of Trapezium Formula Using a Triangle
We will derive the area of a trapezium formula by using a triangle here. Consider the above trapezium of bases a and b and height h. In order to derive the formula,
- Step 1: Split one of the legs of the trapezium into two equal parts.
- Step 2: Cut a triangular portion from the trapezium (as shown in the top figure of the below diagram).
- Step 3: Attach it at the bottom (as shown in the bottom figure in the diagram below).
The trapezium can thus be rearranged as a triangle. It can be concluded from the above diagram that the areas of both the trapezium and the triangle are equal. Also, it can be observed that the base of the triangle is equal to (a + b) and the height of the triangle is h.
The area of the trapezium = The area of the triangle = ½ × base × height = ½ (a + b) h
How To Calculate Area of Trapezium?
The steps given below can be followed to find the area of a trapezium:
- Step 1: Note the dimensions of the lengths of the parallel sides (bases) of the trapezium from the given data.
- Step 2: Calculate the sum of the bases.
- Step 3: Multiply the value of the sum of bases by height or altitude of the trapezium and by 1/2.
- Step 4: Write the answer in square units.
Example: Find the area of a trapezium whose bases are 22 in and 10 in and whose height is 5 in.
The bases are a = 22 in; b = 10 in.
The height is h = 5 in.
The area of the trapezium is, A = ½ (a + b) h
⇒ A = ½ (22 + 10) (5) = ½ (32) (5) = 80 in2.
Let us have a look at a few solved examples to understand the area of trapezium better.
Area of Trapezium Examples
Example 1: Find the height of the trapezium if its area is 128sq.in and the lengths of the bases are 18in and 14in.
Length of bases of trapezium = 18 in and 14 in
Area = 128 sq in
Let the height of the trapezium be 'h' units.
Using the trapezium formula, we know, Area = ½ (Sum of bases) height
Substitute all these values in the area of trapezium formula,
128 = ½ (18 + 14) h
⇒ h = 128 × 2/32
⇒ h = 8 in
Answer: The height of the given trapezium = 8 in.
Example 2: Find the area of an isosceles trapezium with the length of each leg to be 13 units and the bases are of lengths 22 units and 12 units.
Bases, b = 22 units; a = 12 units.
From the above figure,
x + x + 12 = 22
⇒ 2x + 12 = 22
⇒ 2x = 10
⇒ x = 5
Using Pythagoras theorem,
x2 + h2 = 132
⇒ 52 + h2 = 169
⇒ 25 + h2 = 169
⇒ h2 = 144
⇒ h = √144 = 12
The area of the given trapezium is,
A = ½ (a + b) h
⇒ A = ½ (12 + 22) (12) = 204 square units
Answer: The area of the given trapezium = 204 square units.
FAQs on Area of Trapezium
What is the Area of a Trapezium?
The area of a trapezium is the total space covered by the shape in the two-dimensional plane. To calculate the area of a trapezium, we simply multiply the sum of bases with the height and divide the obtained product by 2.
How to Find the Area of Trapezium Without Height?
If the height of the trapezium is not given and all its sides are given instead, then we will divide it into two right triangles and a rectangle. The areas of each of these shapes can be calculated and added to find the area of the given trapezium.
How to Calculate Area of Trapezium Using Base and Height?
The area of a trapezium, given the height and bases, can be calculated by multiplying the average of the base lengths with the height of the trapezium. The height of a trapezium is simply the distance between the parallel sides.
What are the Formulas for Perimeter and Area of Trapezium?
The area of a trapezium can be calculated using the formula: A = ½ × (a + b) × h. While the formula to calculate the perimeter of the trapezium is given as: P = a + b + c + d.
- a, b, c, and d = Length of sides of a trapezium
- h = Distance between the two parallel sides i.e., a and b.
How do you Prove the Area of a Trapezium Formula?
The area of a trapezium formula can be proved by rearranging the trapezium in form of a parallelogram or a triangle. The area of the shapes thus obtained can be related to a trapezium for proving the area of a trapezium formula.
What is the Use of Area of Trapezium Calculator?
Area of trapezium calculator is used to calculate the area of trapezium by entering the given specific parameters. Such as height and the value of the bases. It gives the measurement easily and quickly. Try Cuemath's area of trapezium calculator and get your answers just by a click.
How do you Find the Area of a Trapezium Using Sides?
The area of a trapezium can be calculated if the measurements of its sides are known. For this, we can divide the shape in the form of a rectangle and 2 right-angled triangles. The height can be calculated applying Pythagoras theorem in one of the right-angled triangles and finally, we can apply the area of the trapezium formula to calculate the required area.